A* Shortest Path - Egress Diagram (Almost There But Need Help)

That is true. However, if the mesh is a fairly isotropic triangle mesh (e.g. remeshed using TriRemesh), and you operate on face-face graphs, that does appear to work fairly predictably. See an example here:

Yes it can be pretty tricky to find the right combination of resolution/constraints/settings. Edit: Regarding the cutting corners issue, one might use the Collider goal to prevent the relaxed polyline edges cutting into vertices of the mesh naked perimeter vertices. I haven’t tried it, but it might be worth considering.

The example you posted can be solved using the convex hull to find two paths around the obstacle and picking the shorter one, or using a visibility graph. Which can both be extended to work on 2.5D meshes. It would probably help if we had a mesh/case that can not be solved using such relatively simple methods.